2021/04/30 by Michael Borinsky, Nikhef Theory Group, The Netherlands, Gerald V. Dunne +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebraic number #Hopf algebra #Mathematical analysis #Mathematical physics #Mathematics #Non-perturbative #Nonlinear Waves and Solitons #Physics #Pure mathematics #Quantum #Quantum chaos and dynamical systems #Quantum electrodynamics #Quantum mechanics #Semiclassical physics #Series (stratigraphy) #hep-th #math-ph #math.MP
paper · pdf · doi:10.3842/sigma.2021.087
published as SIGMA 17 (2021), 087, 26 pages
arxiv created 2021/09/23 · openalex publication_date 2021/09/23 · arxiv updated 2021/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We analyze the asymptotically free massless scalar 3 quantum field theory in 6 dimensions, using resurgent asymptotic analysis to find the trans-series solutions which yield the non-perturbative completion of the divergent perturbative solutions to the Kreimer-Connes Hopf-algebraic Dyson-Schwinger equations for the anomalous dimension. This scalar conformal field theory is asymptotically free and has a real Lipatov instanton. In the Hopf-algebraic approach we find a trans-series having an intricate Borel singularity structure, with three distinct but resonant non-perturbative terms, each repeated in an infinite series. These expansions are in terms of the renormalized coupling. The resonant structure leads to powers of logarithmic terms at higher levels of the trans-series, analogous to logarithmic terms arising from interactions between instantons and anti-instantons, but arising from a purely perturbative formalism rather than from a semi-classical analysis.